We use holography to study the large spin J limit of the spectrum of low energy states with charge Q under a U(1) conserved current in CFTs in d>2 dimensions, with a focus on d=3 and d=4. For Q=2, the spectrum of such states is known to be universal and properly captured by the long-distance limit of holographic theories, regardless of whether the CFT itself is holographic. We study in detail the holographic description of such states at Q>2, by considering the contribution to the energies of Q scalar particles coming from single photon and graviton exchange in the bulk of AdS; in some cases, scalar exchange and bulk contact terms are also included. For a range of finite values of Q and J, we numerically diagonalize the Hamiltonian for such states and examine the resulting spectrum and wavefunctions as a function of the dimension Δ of the charge-one operator and the central charges cT, cJ of the stress tensor and U(1) current, finding multiple regions in parameter space with qualitatively different behavior. We discuss the extension of these results to the regime of parametrically large charge Q, as well as to what extent such results are expected to hold universally, beyond the limit of holographic CFTs. We compare our holographic computations to results from the conformal bootstrap for the 3d O(2) model at Q=3 and Q=4 and find excellent agreement. Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_physique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

The study of (large) systems of interacting particles in mathematics is a subject with multiple applications that can be approached from a wide variety of viewpoints and methods. The aim of this day is to bring together young researchers working on similar objects with different tools to discuss and compare methods. It will feature five long presentations discussing recent advances while retaining an introductory focus.Registration is free and open until May 30, 2024.Invited speakers:Thomas Cavallazzi (CentraleSupélec, Paris-Saclay)Louis-Pierre Chaintron (DMA, ENS)Antonin Chodron de Courcel (IHES)Rodrigue Lelotte (CERMICS, ENPC)Sonia Velasco (MAP5, Paris-Cité)Organizers: Laetitia Colombani (CMAP) & Pierre Le Bris (IHES)

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications 

The Seed seminar of mathematics and physics is a seminar series that aims to foster interactions between mathematicians and theoretical physicists, with both online and in-person events. It is holding a special day on Matrix models for quantum systems at IHES, with contributions from Guillaume Aubrun, Philippe Biane, Bertrand Eynard and Vladimir Kazakov. Registration is free and open until May 31, 2024.Invited speakers:Guillaume Aubrun (Institut Camille Jordan, Lyon)Philippe Biane (Laboratoire d’Informatique Gaspard Monge, Marne-la-Vallée)Bertrand Eynard (Institut de Physique Théorique, CEA Saclay)Vladimir Kazakov (Laboratoire de Physique de l’École Normale Supérieure, Paris)Scientific Committee: Thierry Bodineau ( IHES)Slava Rychkov (IHES)Organizing Committee: Ariane Carrance (CMAP)Matteo D’Achille (LMO)Edoardo Lauria (LPENS)

Seminar on Quantum Modularity and ResurgenceThe partition function of topological string theory is an asymptotic series in the topological string coupling and provides in a certain limit a generating function of Gromov-Witten (GW) invariants of a Calabi-Yau threefold. I will discuss how the resurgence analysis of the partition function allows one to extract BPS invariants of the same underlying geometry. I will further discuss how the analytic functions in the topological string coupling obtained by Borel summation admit a dual expansion in the inverse of the topological string coupling leading to another asymptotic series at strong coupling and to the notion of topological string S-duality. This S-duality leads to a new modular structure in the topological string coupling. I will also discuss relations to difference equations and the exact WKB analysis of the mirror geometry. This is based on various joint works with Lotte Hollands, Arpan Saha, Iván Tulli and Jörg Teschner as well as on work in progress.========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

The simplest topologically ordered phase in 2+1D is the deconfined phase of Z2 lattice gauge theory. There are two reasonably well-understood ways to exit the deconfined phase: the Higgs transition, where electric charge (the « e » anyon) condenses, and the confinement transition, where magnetic charge (the « m » anyon) condenses. However, we can also exit the deconfined phase via the self-dual line in the phase diagram, where there is a symmetry between « e » and « m ». What happens here is more mysterious. If this transition is continuous, it may be the simplest critical point with no useful continuum Lagrangian (as yet). After reviewing the formulation of the model as the statistical mechanics of membranes, I will describe clear Monte Carlo evidence for the continuity of the self-dual transition. I will sketch why it is not a conventional « Landau » critical point. Separately, I will use the membrane formulation to describe a very concrete and intuitive way of understanding the emergent higher-form symmetries which appear in part of the phase diagram (and which are the reason that the Higgs and confinement transitions can be understood using Landau theory, despite lacking local order parameters). Work with Andres Somoza and Pablo Serna (https://arxiv.org/abs/2012.15845 and https://arxiv.org/abs/2403.04025). Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_physique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Probability and analysis informal seminarOn a finite connected graph, the product of the non-zero eigenvalues of the Laplacian counts the rooted spanning trees, according to a theorem often attributed to Kirchhoff (1847), or sometimes to Sylvester (1857). Among many generalisations of this classical result, those of Zaslavsky (1982), Forman (1993) and Kenyon (2011) state that when we twist the Laplacian by putting a sign or a phase, complex or quaternionic, on each edge, its determinant counts covering forests of unicycles, with appropriate weights. A common feature of all these results is that the random subgraphs naturally associated to each of these situations (uniform spanning trees and random covering forests of unicycles), seen as random subsets of the (finite) set of edges of the ambient graph, are determinantal point processes. I will present some results of an ongoing joint work with With Adrien Kassel (CNRS, ENS Lyon) in which we investigate further extensions of these results to the covariant Laplacian associated with an arbitrary unitary connection, that is, to the Laplacian twisted by a unitary matrix on each edge. In a first part, I will describe the classical results of Kirchhoff and Forman, then (from a perhaps slightly unorthodox point of view) determinantal point processes on finite sets, and explain what the ones have to do with the others. In a second part, I will describe the measures on Grassmannians that we introduced with Adrien Kassel, explain why they are relevant to the understanding of the twistes Laplacian, and finally describe, to the extent that we understand them, the new random objects that appear over a graph endowed with a unitary connection.========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Seminar on Quantum Modularity and ResurgenceThe talk will focus on quantum modularity relations satisfied by the $q$-Pochhammer symbol  $(q)_N = (1-q) … (1-q^N)$ at $q=exp(2 pi i x)$. These formulas can be interpreted as finite analogues of the usual modularity for the Dedekind eta-function. We’ll discuss certain aspects which come very handy upon summing over $N$. We’ll explain how these can be used, in the context of Kashaev’s invariant of hyperbolic knots, to prove, in a few cases, Zagier’s quantum modularity conjecture by means of what we currently know on the Volume Conjecture. This is based on joint work with Sandro Bettin.========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

The goal of this conference is to advance the dialogue and interactions between the LLM community and the larger world of mathematics in order to further the mathematical understanding of LLMs and contribute to solving some of the outstanding problems in the new field of LLMs. In particular we intend to investigate mathematical structures that can be used to understand LLMs in terms of what they implicitly learn and how.  At the same time, in the opposite direction the use of LLMs in order to do mathematics will be investigated. Registration is free and open until May 16, 2024.Invited speakers:François Charton (Meta AI Research)Andrew Dudzik (Google DeepMind)Amaury Hayat (École des Ponts ParisTech & CERMICS)Julia Kempe (NYU Center for Data Science & CIMS)Gabriel Synnaeve (Meta AI Research)Yiannis Vlassopoulos (Athena Research Center & IHES)Organizers: François Charton (Meta AI Research), Michael Douglas (Harvard University & IHES) & Yiannis Vlassopoulos (Athena Research Center & IHES) 

In my previous talk, on 9/4/24, I set Stark’s Conjectures in the more general context of Hilbert’s 12th Problem, highlighting the special complex functions used by number-theorists to study various cases in recent decades. I also surveyed the remarkable way that the same special functions have cropped up recently in Quantum and Statistical Physics, as indeed have SICs themselves in the case of the first order Stark Conjecture over real quadratic fields.In this second, more number-theoretic, talk I will focus on the latter case. After recalling the necessary details, I will motivate and explain some ongoing work which sets SICs in the context of the Heisenberg group over ${mathbb Z}_p$ (the p-adic integers), `Theta-pairings’ of p-adic measures and Coleman’s power series. This in turn motivates the search for `special measures’ to replace the complex functions mentioned above, in a possible p-adic theory of real-multiplication.Although this will necessarily be a more technical talk than the previous one, I shall still aim to make it largely accessible to non-number-theorists.========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Albert Schwarz started as a topologist/geometer in the early ’50s, then in the ’70s he began an exploration of mathematical aspects of quantum field theory and made numerous seminal contributions to this subject. He has been a regular visitor to IHES since 1995 and was involved in many significant collaborations. This May he will give a short course on his recent research. The mini-conference is a tribute to the 70th anniversary of Schwarz’s remarkable scientific career by several of his friends and colleagues. Registration is free and open until June 7, 2024.Invited speakers:Alain Connes (Collège de France & IHES)Anton Kapustin (Caltech)Maxim Kontsevich (IHES)Boris Pioline (LPTHE – Sorbonne Université)Albert Schwarz (University of California at Davis & IHES)Organizers: Anton Kapustin (Caltech) & Maxim Kontsevich (IHES)